Existence of Energy Minimizers as Stable Knotted Solitons in the Faddeev Model

Fanghua Lin Courant Institute of Mathematical Sciences New York University New York, New York 10021

and

Yisong Yang Department of Mathematics Polytechnic University Brooklyn, New York 11201

Abstract

In this paper, we study the existence of knot-like solitons realized as the energy-minimizing configurations in the Faddeev quantum field theory model. Topologically, these solitons are characterized by an Hopf invariant, Q, which is an integral class in the homotopy group π_{3}(S^{2}) = ℤ. We prove in the full space situation that there exists an infinite subset 𝕊 of ℤ such that for any m ∈ 𝕊, the Faddeev energy, E, has a minimizer among the topological class Q = m. Besides, we show that there always exists a least-positive-energy Faddeev soliton of non-zero Hopf invariant. In the bounded domain situation, we show that the existence of an energy minimizer holds for 𝕊 = ℤ. As a by-product, we obtain an important technical result which says that E and Q satisfy the sublinear inequality E ≤ C|Q|^{3/4}, where C > 0 is a universal constant. Such a fact explains why knotted (clustered soliton) configurations are preferred over widely separated unknotted (multisoliton) configurations when |Q| is sufficiently large.

1 Introduction

The understanding of solution structures of the governing equations of classical or quantum field theory models often leads physicists to discovering revolutionary concepts and making new theoretical advances. Among the well-known examples are blackholes, monopoles, vortices, and instantons. For a long time, elementary particles and many other elementary structures are visualized as point-particles, which makes the study of point-like solitons of prominent importance and a broad range of results concerning their existence, behavior, construction, classification, quantumtheoretical interpretations, etc, have been obtained.

The impetus of the study of knots came when Lord Kelvin explored the idea that atoms could be thought of made of knotted vortex tubes of ether so that the stability of matter could be explained by the topological stability of knots, the variety of chemical elements could be viewed as a consequence of the variety of knots, and the spectral lines of atoms could be considered as a reflection of the oscillatory patterns of knots [?, ?]. In the modern era of science, the idea of knots has inspired various fundamental areas, including particle and condensed-matter physics, statistical mechanics, cosmology, molecular biology, and synthetic chemistry. For example, knots may be used to explain the concept of spin [?]; elementary particles may be regarded as quantized flux loops represented by knots or links and antiparticles by their mirror images [?]; knotted cosmic strings may be produced in the early stages of the universe which are responsible for the initial matter accretion for galaxy formation [?, ?, ?, ?]; knotted structures may appear in ferromagnetic spin-triplet superconductors [?]; the topological classification of knots may give clues to various aspects of DNA [?]; the entanglement structures of polymers may also be understood based on a knowledge on knots [?].

Numerous contributions to the classification of knots have been made. Notably, Tait [?] enumerated knots in terms of the crossing number of a plane projection; Alexander [?] discovered a knot invariant, known as the Alexander polynomial, arising in 3-dimensional homology; Jones [?, ?] found a new knot invariant, known as the Jones polynomial, which enabled several conjectures of Tait to be proved [?, ?]; based on a heuristic quantum-field theory argument, Witten [?] derived from the Chern–Simons action a family of knot invariants including the Jones invariant; finally came the Vassiliev invariants [?] which cover the Alexander polynomial and the Jones polynomial and lay a general framework for the study of the combinatorial aspects of knots.

Despite the remarkable progress in the understanding of the topology and combinatorics of knots, the study on the dynamics of knots as the solution configurations of suitable Lagrangian field-theory equations has only recently been brought into light by Faddeev and Niemi [?, ?, ?] who used computer simulation, relaxation and toroidal coordinates to show that a ring-shaped (unknotted) Hopf charge one soliton exists as the energy minimizer of a relativistic quantum field theory model proposed many years ago by Faddeev [?]. Shortly after the seminal work of Faddeev and Niemi, a more extensive computer investigation was conducted by Battye and Sutcliffe [?, ?, ?] who performed fully three-dimensional, highly convincing, computations for the solution configurations of the Hopf charge Q from Q = 1 up to Q = 8 and found that, for Q = 1, 2, 3, 4, 5, the energy-minimizing solitons are ring-shaped and higher charges cause greater distortion, and for Q = 6, 7, 8, the solitons become knotted and higher charges cause greater complexity. See also [?]. Thus, it has become imperative to obtain a rigorous proof for the existence of these knotted solitons as the energyminimizing configurations of the Faddeev model. The aim of the present paper is to develop such an existence theory.

Here is an outline of the rest of the contents of the paper. In §2, we review the Faddeev model and its topological characterization and state the problem to be studied. We then state our main existence theorems and make some due comments about these results in the perspectives of general field theory models. In §3, we study the integral representation of the Hopf invariant in view of the natural mapping space (admissible class) and we justify that the integral may and may only assume integer values. In particular, we show that this invariant is preserved under the weak convergence induced by the Faddeev energy functional. In §4, we establish our existence theorem in the case when the underlying domain is bounded. We shall see that an energy minimizer exists among any topological class. In §5, we establish our important preliminary lemma that the Faddeev energy obeys a sublinear upper estimate in terms of the topological invariant according to a 3/4-power law. Such a result complements the lower energy estimate of Vakulenko and Kapitanski [?] and explains in particular why, for configurations with high Hopf invariants, locally clustered solitons are preferred over multicentered solitons. In §6, we start our study of the convergence (compactness) problem over the entire space ℝ^{3} for an energy minimizing sequence and we establish an additivity result for the energy infima evaluated over a given topological class and its decomposed (evading) unknown topological subclasses along the minimization process. In §7, we extend our previous analysis to several situations. In §7.1, we prove that the minimum positive-energy minimizers (or the least-mass Faddeev solitons) exist. Furthermore, as an essential application of our sublinear energy growth law obtained in §5, we show that there exists an unknown infinite subset 𝕊 of the set of all integers ℤ so that there exists a solution for the energy minimization problem over each topological class defined by any number in 𝕊. In §7.2, we revisit the Skyrme model and prove the existence of an energy minimizer among all unit topological degree field configurations. In §7.3, we study the minimization process in view of a technical lemma and prove that the weak limits attained in §6 among the decomposed topological subclasses given by various corresponding Hopf indices in Theorem ?? are in fact the solutions to their respective minimization problems characterized by those topological indices. As a by-product, we shall also prove an energy-additivity relation. In §8, we summarize the paper.

Technically, since our main concern is about deriving useful consequences from an energy-minimizing sequence of an energy functional over the full space ℝ^{3}, we encounter familiar difficulties in this type of analysis. In fact, the concentrationcompactness principle of Lions [?, ?] says that, in this situation, one may have just one possibility among (i) compactness (concentration), (ii) vanishing, and (iii) dichotomy, and for the solution of a minimization problem, the best one would be able to achieve is (i), of course. For the Faddeev model, however, (i) remains elusive for us. Indeed, we shall show that (ii) cannot occur and that a combined form of (i) plus (iii) leads to our main existence theorem. We shall see that due to the topological constraint in the minimization problem, any energy-minimizing sequence contains a subsequence that will concentrate around a (uniformly) finite number of repulsive local regions in the limiting process and that these local regions absorb “additively decomposed” topology assigned to the minimizing sequence. Hence, in the subsequence limit, we obtain the solutions of the minimization problem over the topological classes of these “additively decomposed topological components” of the original class. Using the sublinear growth estimate of the energy with respect to the topological charge or the Hopf invariant, we are able to deduce that the minimization problem of the Faddeev model can be solved when the Hopf invariant assumes its value in an infinite subset of the set of all integers.

It should be pointed out that the complexity of the structure of the problem has not allowed us to study the fine geometry of the obtained energy minimizers or the question of how knotted these solutions are.

For a broader readership and consistency with current literature, we use in §2 tensor notation to introduce the problem and state the main existence results. For convenience, starting from §3 (except in §7.2 when we study the Skyrme model), we use differential forms in our mathematical analysis of the problem. Such a transition should be natural for a mathematics oriented reader.

2 Main Results

Recall that, in normalized form, the action density of the Faddeev model [?, ?, ?, ?, ?] over the standard (3 + 1)-dimensional Minkowski space of signature (+ − −−) reads

L = ∂_{μ}n · ∂^{μ}n − \frac{1}{2}F_{μν}(n)F^{μν}(n),  (2.1)

where the field n = (n_{1}, n_{2}, n_{3}) assumes its values in the unit 2-sphere, i.e., n^{2} = n_{1}^{2} + n_{2}^{2} + n_{3}^{2} = 1, and

F_{μν}(n) = n · (∂_{μ}n ∧ ∂_{ν}n).  (2.2)

Since n is parallel to ∂_{μ}n ∧ ∂_{ν}n, it is seen that

F_{μν}(n)F^{μν}(n) = (∂_{μ}n ∧ ∂_{ν}n) · (∂^{μ}n ∧ ∂^{ν}n),  (2.3)

which may be identified with the well-known Skyrme term [?, ?, ?, ?, ?] when one embeds S^{2} into S^{3} ≈ SU(2). Hence, as observed by Cho [?], the Faddeev model may be viewed as a refined Skyrme model and the solution configurations of the former are the solution configurations of the latter with a restrained range. In what follows, we shall only be interested in static fields which make the Faddeev energy

E(n) = ∫_{ℝ^{3}} { ∑_{1≤k≤3} |∂_{k}n|^{2} + ∑_{1≤k<ℓ≤3} F_{kℓ}^{2}(n)} dx  (2.4)

finite. The finite-energy condition implies that n approaches a constant vector n_{∞} at spatial infinity (of ℝ^{3}). Hence we may compactify ℝ^{3} into S^{3} and view the fields as maps from S^{3} to S^{2}. As a consequence, we see that each finite-energy field configuration n is associated with an integer, Q(n), in π_{3}(S^{2}) = ℤ. In fact, such an integer Q(n) is known as the Hopf invariant which has the following integral characterization: The differential form F = F_{kℓ}(n)dx^{k} ∧ dx^{ℓ} (k, ℓ = 1, 2, 3) is closed in ℝ^{3}. Thus, there is a one form, A = A_{k}dx^{k} so that F = dA. Then the Hopf charge Q(n) of the map n may be evaluated by the integral

Q(n) = \frac{1}{16π^{2}} ∫_{ℝ^{3}} A ∧ F,  (2.5)

which is correctly a Chern–Simons index. Alternatively, we may use the vector notation, F(n) = (F_{k}), F_{k} = \frac{1}{2}ϵ_{kℓℓ′}F_{ℓℓ′}(n) (the Hodge dual). Then F(n) is divergence free, ∇ · F(n) = 0. So there is a vector field A such that F(n) = ∇ × A and the Hopf invariant Q(n) has the expression

Q(n) = \frac{1}{4π^{2}} ∫_{ℝ^{3}} A · F(n) dx.  (2.6)

Of course, the classical field configurations of the Faddeev model are the solutions to the problem

E_{m} = inf{E(n) | E(n) < ∞, Q(n) = m},  (2.7)

where m is an integer. As mentioned earlier, the computer simulations in [?, ?, ?, ?] are for m = 1, 2, ⋯, 8 for which the problem (??) is truncated over a finite large box which “approximates” the full space ℝ^{3}.

Here are our main existence theorems for the fundamental problem (??).

(i) The case of the entire space

The existence of a least-energy Faddeev soliton

Note that the lower estimate ensures the existence of an integer m_{0} ≠ 0 such that

E_{m_{0}} = min{E_{m} | m ∈ ℤ \ {0}}.

For such m_{0}, we have

Theorem 2.1. The problem (??) with m = m_{0} has a solution.

The existence of energy minimizers among infinitely many topological classes Using Theorem ?? and the sublinear energy growth law (see (??)) to be derived

in §5, we have

Theorem 2.2. There exists an infinite subset 𝕊 of the set ℤ of all integers so that for any m ∈ 𝕊 the problem (??) has a solution.

Although we do not know how big the set 𝕊 stated in Theorem ?? is or whether 𝕊 ≠ ℤ, the following theorem indicates that 𝕊 is quite substantial:

Theorem 2.3. For any m ∈ ℤ, there is a decomposition

m = m_{1} + m_{2} + ⋯ + m_{ℓ}, m_{s} ∈ 𝕊, s = 1, 2, ⋯, ℓ,  (2.8)

so that the following sub-additivity relation

E_{m} ≥ E_{m_{1}} + E_{m_{2}} + ⋯ + E_{m_{ℓ}}  (2.9)

holds. Furthermore, the sublinear growth (upper) estimate

E_{m} ≤ C_{2}|m|^{3/4}  (2.10)

is valid. Here, in (??) and (??), C_{1} and C_{2} are universal positive constants.

Note that the lower bound

E(n) ≥ C|m|^{3/4}  (2.11)

was derived a long time ago by Vakulenko and Kapitanski [?]. Hence we see that the sharp sublinear growth estimate E_{m} ∼ |m|^{3/4} holds asymptotically for a large Hopf charge |m|.

(ii) The case of a bounded domain

When we consider the problem over a bounded contractible domain, a more satisfactory result is valid because, technically, a bounded domain prohibits the minimizing sequence to “float” away and “concentration” is trivially guaranteed:

Theorem 2.4. Let Ω be a bounded contractible domain in ℝ^{3} and consider the admissible set of all the field configurations which assume a constant value on the boundary of and outside Ω. Then, over such an admissible set, the problem (??) has a solution for any m ∈ ℤ.

This theorem ensures the existence of the knotted solutions of respective Hopf charges obtained in [?, ?, ?, ?, ?] where the full space ℝ^{3} is replaced by a large box in order to carry out computer simulations.

In view of a technical statement (Lemma ??), we can strengthen Theorem ?? as follows:

Theorem 2.5. For any m ∈ ℤ, let {n_{i}} be a minimizing sequence of the problem (??). Then, there is subsequence which is still denoted by {n_{i}} for convenience and a decomposition

m = m_{1} + m_{2} + ⋯ + m_{ℓ}, 1 ≤ ℓ ≤ C_{1}|m|^{3/4},  (2.12)

where m_{1}, m_{2}, ⋯, m_{ℓ} ∈ 𝕊 so that one can always find correspondingly suitably modified subsequences {n_{(1,i)}}, {n_{(2,i)}}, ⋯, {n_{(ℓ,i)}} from {n_{i}} which converge in the weak topology over ℝ^{3} induced by the Faddeev energy to the respective solutions of the minimization problems

E_{m_{s}} = inf{E(n) | E(n) < ∞, Q(n) = m_{s}}, s = 1, 2, ⋯, ℓ.  (2.13)

Besides, there holds the additivity relation

E_{m} = lim inf_{i→∞} E(n_{i}) = E_{m_{1}} + E_{m_{2}} + ⋯ + E_{m_{ℓ}}.  (2.14)

It is easy to see that Theorem ?? also implies Theorem ??. In fact, if 𝕊 is finite, then there is a positive integer q so that |m_{s}| ≤ q for any m_{s} ∈ 𝕊. Using (??), we have

|m| ≤ qℓ ≤ qC_{1}|m|^{3/4},

which is impossible when |m| is sufficiently large.

The equality (??) says that the full spectrum of the Faddeev soliton masses, {E_{m} | m ∈ ℤ }, may be “generated” by the subclass masses, {E_{m} | m ∈ 𝕊 }.

We note that a profound implication of (??) is indeed the existence of knotted solitons at sufficiently high Hopf charges. To see this, we show that, for a large value of Q, a Faddeev energy minimizer prefers to appear as a clustered configuration (a knotted soliton) realizing the topology designated by Q than appear as a field configuration with widely separated energy lumps of a simpler topology (a multisoliton of a sum of unknotted solitons) realizing the same topology. Such a result may be illustrated most easily by showing that, if m > 0 is sufficiently large, a Faddeev energy minimizer with the Hopf invariant Q = m can never be represented as a multisoliton (say, for simplicity) of the sum of m widely separated solitons, each of a Hopf charge Q = 1 (an unknot). In fact, let the value of (??) be denoted by e_{m}. If the above described multisolitons were allowed, then, away from the local concentration regions of these unknots, the field configurations gave negligible contributions to the total energy. Hence, approximately, we would have e_{m} ≈ me_{1}, which contradicts (??) for large m.

We remark that the |m|^{3/4} asymptotic law was also surprisingly proven by Riviere [?], for a conformally invariant variational problem for maps from S^{3} into S^{2}. There, he obtained similar results, though for a rather different problem defined on a compact domain.

Significant difficulties arise when we attempt to gain further knowledge about the set 𝕊 stated in Theorem ?? because a minimizing sequence of the problem (??) may fail to “concentrate” in ℝ^{3}.

It will be of interest to comment on the importance of Theorem ?? and Theorem ?? with regard to the general existence problem for topological solitons in quantum field theory. Recall that Belavin and Polyakov [?] were able to construct all static solitons characterized by an arbitrary topological charge (the Brouwer degree) for the σ-model modeling the spin vector orientation for a planar ferromagnet. The four-dimensional extension of this construction is of course the well-known resolution [?] of the classical Yang–Mills instantons realizing again any prescribed topological charge (the second Chern number). The common feature of these two soluble models is that they are both conformally invariant field theories. When conformal invariance becomes invalid, the above-described complete solvability may not be available. For example, except in the critical phase [?] between two types of superconductivity, people have not been able to establish for the Ginzburg–Landau theory on ℝ^{2} the existence of an energy minimizer realizing a given quantized flux (the first Chern number), and a similar situation happens for the Chern–Simons theory [?, ?]; except in the BPS limit [?, ?, ?], people have not been able to establish the existence of a Yang–Mills–Higgs monopole of any monopole number (the winding number); although there have been some works on the existence of energy-minimizing unit-charge Skyrme solitons [?, ?, ?, ?, ?], the proofs seem problematic unfortunately (in §7.2, we give a proof of the existence of degree one Skyrme solitons to fill this gap). Thus, in view of the above picture with regard to the role played by conformal invariance in the solvability of quantum field theory models, it should not come as a great surprise to see that Aratyn, Ferreira, and Zimmerman [?] were able to obtain in toroidal coordinates a wide family of explicit solutions realizing all possible integer values of the Hopf invariant for their curiously defined field theory model with a static Hamiltonian density of the form

H = ∑_{1≤k<ℓ≤3}(F_{kℓ}^{2}(n))^{3/4},  (2.15)

which is conformally invariant over ℝ^{3}. However, the original Faddeev energy (??) is not conformally invariant and is more difficult.

Remark. It will be interesting to know the value of m_{0} in Theorem ??. Using the Hopf map H : S^{3} → S^{2} (see §3.3) composed with an inverse stereographic projection, Ward [?] found that E_{1} ≤ 32\sqrt{2}π^{2}. Besides, the constant C in (??) may be estimated [?] to yield a lower bound C ≥ 8\sqrt{2}3^{3/8}π^{2}. Consequently, E_{m} > E_{1} whenever |m| > 3, which implies that m_{0} can only be ±1, ±2, or ±3. On the other hand, according to the numerical work of Battye and Sutcliffe [?, ?, ?], E_{1} < E_{m} for m = 2, 3, ⋯, 8. Therefore, it is fairly safe to assert that m_{0} = ±1.

3 Mapping Spaces

In this section, we study the mapping spaces naturally associated with the Faddeev energy functional. In §3.1, we reformulate the problem in terms of differential forms and introduce a convenient “gauge” condition and an induced weak topology. In §3.2, we consider two preliminary layers, called X_{1} and X_{2}, of the function space (admissible class) and study the Hopf invariant. In §3.3, we prove that the Hopf invariant assumes integer values over the entire X_{2}. In §3.4, we show that X_{2} is the most natural admissible class for the minimization problem of the Faddeev model. The study here lays an analytical foundation for our problem.

3.1 Hopf invariant for maps from S^{3} to S^{2}

It will be convenient to recall the Hopf invariant of a smooth map u from S^{3} to S^{2}.

Let ω_{S^{2}} be the canonical area form of S^{2}. The pull-back of ω_{S^{2}} under u, u^{#}(ω_{S^{2}}), is a closed 2-form on S^{3}. Since the second de Rham cohomology on S^{3} is trivial, HdR^{2}(S^{3}, ℝ) = {0}, there is a 1-form η on S^{3} such that dη = u^{#}(ω_{S^{2}}). Then the Hopf invariant that defines the integer class of the map u : S^{3} → S^{2} as an element of the homotopy space H_{3}(S^{2}) = ℤ is given by the integral

Q(u) = \frac{1}{16π^{2}} ∫_{S^{3}} η ∧ u^{#}(ω_{S^{2}}).  (3.1)

Note that the above-mentioned η is unique up to a closed 1-form. Since HdR^{1}(S^{3}) = {0}, any two possible such η’s can only differ by an exact 1-form. Hence, in view of the Stokes theorem, the value of Q(u) is independent of the choice of η. In fact, one can easily verify that the value of Q(u) is also independent of the choice of a particular 2-form on S^{2}. In other words, the canonical area element ω_{S^{2}} in the definition of the Hopf invariant Q(u) may be replaced by any 2-form ω on S^{2} satisfying

\frac{1}{4π} ∫_{S^{2}} ω = 1.  (3.2)

See [?] for more details. For any map u : ℝ^{3} → S^{2} such that u is of class C^{1} and such that u(x) goes to a definite limit sufficiently fast as |x| → ∞, u may be viewed as a map from S^{3} to S^{2}. Since (??) is independent of the choice of coordinates on S^{3}, the Hopf invariant of u becomes

Q(u) = \frac{1}{16π^{2}} ∫_{ℝ^{3}} η ∧ u^{#}(ω_{S^{2}}),  (3.3)

which coincides with (??). With the new notation, the (renormalized) Faddeev energy for u : ℝ^{3} → S^{2} assumes the form

E(u) = ∫_{ℝ^{3}} {|∇u|^{2} + ∑_{1≤k<ℓ≤3} |∂_{k}u ∧ ∂_{ℓ}u|^{2}} dx.  (3.4)

For definiteness, let B be the completion of the set of C^{1}-maps from ℝ^{3} to S^{2} which assume the value (0, 0, 1) ∈ S^{2} near infinity (that is, u(x) = (0, 0, 1) for |x| > R where R > 0 is sufficiently large) with respect to the distance d_{E} induced from the Faddeev energy (??):

d_{E}^{2}(u, v) = ∫_{ℝ^{3}} {|∇(u − v)|^{2} + ∑_{1≤k<ℓ≤3} |∂_{k}u ∧ ∂_{ℓ}u − ∂_{k}v ∧ ∂_{ℓ}v|^{2}} dx.  (3.5)

It can be checked that the metric space (B, d_{E}) is complete.

Proposition 3.1. For each u ∈ B, the Hopf integral invariant Q(u) defined in (??) is well defined and is an integer.

Proof. For any such u, there is a sequence {u_{i}} ⊂ C^{1}(ℝ^{3}, S^{2}) such that u_{i} → u in the metric d_{E} as i → ∞ and u_{i}(x) ≡ (0, 0, 1) for |x| ≥ R_{i} for some R_{i} > 0, i = 1, 2, · · ·. For each u_{i}, we can find a 1-form η_{i} such that

dη_{i} = u_{i}^{#}(ω_{S^{2}}), δη_{i} = 0,  (3.6)

where δ is the adjoint operator of d. In fact, such an η_{i} can be constructed explicitly by the formula

η_{i} = δ( − \frac{1}{4π|x|} ⋆ (u_{i}^{#}(ω_{S^{2}}))),  (3.7)

where ⋆ denotes the convolution.

One easily deduces (see also the discussion below) that, if u_{j} → u in d_{E}, then u_{i}^{#}(ω_{S^{2}}) → u^{#}(ω_{S^{2}}) in both L^{1}(ℝ^{3}) and L^{2}(ℝ^{3}) as i → ∞. In particular, the sequence {∥u_{i}^{#}(ω_{S^{2}})∥_{L^{p}(ℝ^{3})}} remains uniformly bounded for 1 ≤ p ≤ 2. By elliptic estimates, we see that {∥η_{i}∥_{L^{q}(ℝ^{3})}} is also uniformly bounded and precompact in Lloc^{q}(ℝ^{3}) for 1 < q ≤ 6. By (??), η_{i}, in fact, converges as i → ∞ to

η = δ( − \frac{1}{4π|x|} ⋆ (u^{#}(ω_{S^{2}}))), dη = u^{#}(ω_{S^{2}}), δη = 0.  (3.8)

Thus Q(u_{i}) converges as i → ∞ to (??) which is of integer value because Q(u_{i}) are all of integer values. This completes the proof.

3.2 Spaces X_{1} and X_{2}

Let X_{1} be the set of all such maps, u, from ℝ^{3} into S^{2} that there is a sequence {u_{i}} ⊂ C^{1}(ℝ^{3}, S^{2}) with u_{i}(x) = (0, 0, 1) for x near infinity so that

u_{i} ⇀^{d_{E}} u as i → ∞.  (3.9)

That is, E(u_{i}) ≤ C for all i where C > 0 is a constant and that u_{i} → u in L_{loc}^{2}(ℝ^{3}). We remark that X_{1} may not be d_{E}-weakly sequentially closed. The difficulty is the following. Let {u_{i}} ⊂ X_{1} and satisfy (??). Then, for each i, there is a sequence {u_{i,j}} in C^{1}(ℝ^{3}, S^{2}) such that u_{i,j} ⇀^{d_{E}} u_{i} as j → ∞. It is easy to check (by taking a subsequence if necessary) that

E(u_{i,j}) → E(u_{i}) + ν_{i}(ℝ^{3}) as j → ∞,

where ν_{i} is a Radon measure on ℝ^{3}. However, our difficulty is that sup_{i} ν_{i}(ℝ^{3}) may be infinite.

Nonetheless, for each u, one may define the (generalized) Hopf invariant. In fact, for u ∈ X_{1}, one may find a sequence {u_{i}} ⊂ C^{1}(ℝ^{3}, S^{2}) so that (??) is satisfied. Then, since du_{i}^{#}(ω_{S^{2}}) = 0, there is a well-defined 1-form η_{i} on ℝ^{3} satisfying (??) and (??) on ℝ^{3}. The sequence of 2-forms {u_{i}^{#}(ω_{S^{2}})} over ℝ^{3} may be identified in the standard way with a sequence of 3-vectors {V_{i}} over ℝ^{3} and du_{i}^{#}(ω_{S^{2}}) = 0 is equivalent to the divergence-free condition, ∇ · V_{i} = 0. Because u_{i} converges weakly in the metric d_{E}, {V_{i}} is bounded in L^{2}(ℝ^{3}, ℝ^{3}). Since the space

{V ∈ L^{2}(ℝ^{3}, ℝ^{3}) | ∇ · V = 0 in the sense of distributions}

is a Hilbert space (see [?]), one sees that V_{i}⇀V in L^{2}(ℝ^{3}, ℝ^{3}) as i → ∞ and ∇·V = 0. On the other hand, by the weak continuity of the Jacobians (cf. [?, ?]) and (??), one has

∂_{k}u_{i} ∧ ∂_{ℓ}u_{i} ⇀ ∂_{k}u ∧ ∂_{ℓ}u as i → ∞,

for k, ℓ = 1, 2, 3, in the sense of distributions. Since {∂_{k}u_{i} ∧ ∂_{ℓ}u_{i}}_{i} remains uniformly bounded (with respect to i) in L^{2}(ℝ^{3}, ℝ^{3}), we also conclude that ∂_{k}u_{i}∧∂_{ℓ}u_{i} ⇀ ∂_{k}u∧∂_{ℓ}u weakly in L^{2}(ℝ^{3}, ℝ^{3}). By taking subsequence if needed, we may also assume that u_{i} → u a. e. in ℝ^{3}. We thus deduce that u_{i}^{#}(ω_{S^{2}}) ⇀ u^{#}(ω_{S^{2}}) in the sense of distribution, too. Indeed, the coefficients of u_{i}^{#}(ω_{S^{2}}) are of the form u_{i} · ∂_{k}u_{i} ∧ ∂_{ℓ}u_{i}. For any ϕ ∈ C_{0}(ℝ^{3}), there holds

| ∫_{ℝ^{3}} ϕ(u_{i} · ∂_{k}u_{i} ∧ ∂_{ℓ}u_{i} − u · ∂_{k}u ∧ ∂_{ℓ}u) dx|
≤ | ∫_{ℝ^{3}} ϕu · (∂_{k}u_{i} ∧ ∂_{ℓ}u_{i} − ∂_{k}u ∧ ∂_{ℓ}u) dx| + ∫_{ℝ^{3}} |ϕ||u_{i} − u||∂_{k}u_{i} ∧ ∂_{ℓ}u_{i}| dx.

Since ∂_{k}u_{i} ∧ ∂_{ℓ}u_{i} ⇀^{L^{2}} ∂_{k}u ∧ ∂_{ℓ}u as i → ∞, the first term on the right-hand side above goes to zero as i → ∞. That the second term there goes to zero follows from the L^{2}-uniform boundedness of {∂_{k}u_{i} ∧ ∂_{ℓ}u_{i}}_{i}, the fact that u_{i} → u a. e. in ℝ^{3}, and the Lebesgue dominated convergence theorem.

The 3-vector associated with u^{#}(ω_{S^{2}}) must be V as we have described above. Thus d(u^{#}(ω_{S^{2}})) = 0.

On the other hand, elliptic estimates may be used to show that {η_{i}} defined in (??) and (??) is bounded in H^{1}(ℝ^{3}) ∩ L^{q}(ℝ^{3}) uniformly for 3/2 < q ≤ 6. Again by taking a subsequence if needed, we may assume that η_{i} ⇀ η in H^{1}(ℝ^{3}) (weakly) and η_{i} → η in L_{loc}^{q}(ℝ^{3}) (strongly) as i → ∞, 3/2 < q < 6. It is then easy to see that η satisfies (??). (??) also implies that η(x) → 0 as |x| → ∞.

We can thus define Q(u) by (??).

At this point, we do not know whether Q(u) is an integer, for u ∈ X_{1}. If all the u_{i}’s are such that u_{i}(x) ≡ constant for x outside a bounded domain, then it is obvious from our argument that Q(u_{i}) → Q(u) as i → ∞ and, so Q(u) ∈ ℤ.

Let

X_{2} = {u ∈ L^{1}(ℝ^{3}, S^{2}) | E(u) < ∞, d(u^{#}(ω_{S^{2}})) = 0 in the sense of distributions}.

From the above discussion, we immediately conclude that B ⊂ X_{1} ⊂ X_{2}. Moreover, for any u ∈ X_{2}, the (generalized) Hopf invariant Q(u) by (??) is well defined.

3.3 Hopf invariant as a topological degree

We first recall that the Hopf map H : S^{3} → S^{2} may be defined explicitly by the formula H(z, w) = z/w where we identify S^{3} with {(z, w) ∈ ℂ^{2} | |z|^{2} + |w|^{2} = 1} and S^{2} with ℂ ∪ {∞}, the extended complex plane, under the stereographic projection. It is easy to see that H(z, w) = H(z′, w′) if and only if

(z, w) = e^{iθ}(z′, w′) for some θ ∈ ℝ.  (3.10)

For any smooth map u : ℝ^{3} → S^{2} such that u(x) → constant as |x| → ∞, there is a smooth map \overline{u} : ℝ^{3} → S^{3} such that

H ∘ \overline{u} = u.  (3.11)

The existence of such \overline{u} follows from the fact that S^{2} is simply connected. By (??), one sees that such \overline{u} is not unique.

Let ξ be a 1-form on S^{3} such that

dξ = H^{#}(ω_{S^{2}}).

One might also assume that δξ = 0 on S^{3} though it is not needed here. Then

1 = \frac{1}{16π^{2}} ∫_{S^{3}} ξ ∧ H^{#}(ω_{S^{2}})  (3.12)

as the Hopf map has the Hopf invariant 1.

Let us evaluate the Hopf invariant of u by (??) where η is any 1-form on ℝ^{3} such that δη = 0 on ℝ^{3} and η = 0 at infinity (say η is given by (??)). Since

u^{#}(ω_{S^{2}}) = \overline{u}^{#}(H^{#}(ω_{S^{2}})) = \overline{u}^{#}(dξ) = d(\overline{u}^{#}(ξ)),  (3.13)

we might simply choose η = \overline{u}^{#}(ξ). With this choice, we have

Q(u) = \frac{1}{16π^{2}} ∫_{ℝ^{3}} \overline{u}^{#}(ξ) ∧ \overline{u}^{#}(H^{#}(ω_{S^{2}}))
= \frac{1}{16π^{2}} ∫_{ℝ^{3}} \overline{u}^{#}(ξ ∧ H^{#}(ω_{S^{2}})).  (3.14)

In view of (??), we see that Q(u) is simply the topological degree of the map \overline{u} : ℝ^{3} → S^{3} (note that \overline{u} takes a constant value at infinity).

Let u : ℝ^{3} → S^{2} and \overline{u} : ℝ^{3} → S^{3} be as defined in (??). Suppose that dη = u^{#}(ω_{S^{2}}). Then a simple fact proved in [?] is that

|∇\overline{u}|^{2} = \frac{1}{4}|η|^{2} + |∇u|^{2}.  (3.15)

Here we want to generalize these formulas to the case u ∈ X_{2}. We note that u ∈ X_{2} implies in particular that ∫_{ℝ^{3}} |∇u|^{2} dx < ∞ and that d(u^{#}(ω_{S^{2}})) = 0. Hence, by a theorem of Bethuel [?], one may find a sequence {u_{i}} ⊂ C^{1}(ℝ^{3}, S^{2}) such that u_{i} → u in H_{loc}^{1}(ℝ^{3}, S^{2}) as i → ∞. We claim that one may choose such a sequence {u_{i}} that each u_{i} is constant near infinity and that ∇u_{i} → ∇u in L^{2}(ℝ^{3}) as i → ∞.

Indeed, for u : ℝ^{3} → S^{2} with ∫_{ℝ^{3}} |∇u|^{2} dx < ∞, we may choose a sequence of real numbers R_{i} → ∞ as i → ∞ such that

∫_{∂B_{R_{i}}} |∇u|^{2} dσ ≤ \frac{4}{R_{i}} ,

where B_{R} is the ball {x ∈ ℝ^{3} | |x| < R}. Let ξ_{i} denote the average of u over ∂B_{R_{i}},

ξ_{i} = − ∫ _{∂B_{R_{i}}}u ≡ \frac{1}{|∂B_{R_{i}}|} ∫ u dσ.

Then

− ∫ _{∂B_{R_{i}}}|u − ξ_{i}|^{2} ≤ ∫_{∂B_{R_{i}}} |∇u|^{2} dσ → 0

as i → ∞. Thus ξ_{i} → S^{2} as i → ∞.

Consider the problem

ΔV_{i} = 0 in B_{2R_{i}} \ B_{R_{i}}; V_{i} = u on ∂B_{R_{i}}, V_{i} = \overline{ξ}_{i} = \frac{ξ_{i}}{|ξ_{i}|} on ∂B_{2R_{i}}. (3.16)

Later, we shall prove that dist(V_{i}(x), S^{2}) → 0 as i → ∞ for every x ∈ B_{2R_{i}} \ B_{R_{i}}. Moreover,

∫_{B_{2R_{i}}\B_{R_{i}}} |∇V_{i}|^{2} dx ≤ C_{0} ∫_{∂B_{R_{i}}} |∇u|^{2}dx.  (3.17)

From Bethuel’s theorem, one may find u_{i} (for each i) such that u_{i} = \overline{ξ}_{i} on ∂B_{2R_{i}} and u_{i} ∈ C^{1}(B_{2R_{i}}, S^{2}) with

∫_{B_{2R_{i}}} |∇ũ_{i} − ∇u_{i}|^{2} dx = ε_{i} → 0 as i → ∞,

where

ũ_{i}(x) = \{ \begin{matrix}u(x), & x ∈ B_{R_{i}}, \\ \frac{V_{i}(x)}{|V_{i}(x)|}, & x ∈ B_{2R_{i}} \ B_{R_{i}}. \end{matrix}

It is then rather easy to modify u_{i} so that u_{i} ∈ C^{1}(ℝ^{3}, S^{2}), u_{i}(x) =constant for |x| ≥ 3R_{i}, and ∇u_{i} → ∇u in L^{2}(ℝ^{3}) as i → ∞.

Let us now assume u ∈ X_{2}. By the above argument, we may find a sequence {u_{i}} ⊂ C^{1}(ℝ^{3}, S^{2}) such that u_{i}(x) =constant for |x| sufficiently large and that u_{i} → u in H^{1}(ℝ^{3}, S^{2}) as i → ∞.

For each u_{i}, we may consider the corresponding lift \overline{u}_{i} (see (??)) such that (??) holds,

|∇\overline{u}_{i}|^{2} = \frac{1}{4}|η_{i}|^{2} + |∇u_{i}|^{2},  (3.18)

where η_{i} is given in (??). By the relation u_{i} = H ∘ \overline{u}_{i}, one concludes that, as i → ∞, \overline{u}_{i}(x) → \overline{u}(x) for a.e. x ∈ ℝ^{3}, with H ∘ \overline{u}(x) = u(x). Moreover, by the strong convergence of {u_{i}} in H^{1} to u,

η_{i} = δ( − \frac{1}{4π|x|} ⋆ (u_{i}^{#}(ω_{S^{2}}))) → δ( − \frac{1}{4π|x|} ⋆ (u^{#}(ω_{S^{2}}))) = η  (3.19)

in L_{loc}^{q}(ℝ^{3}) for 1 ≤ q < 3/2. This last statement follows easily from the classical potential estimates. Note that δη = 0 and dη = u^{#}(ω_{S^{2}}) with η = 0 at infinity, and that η belongs to better spaces. Indeed, |∇η| ∈ L^{p}(ℝ^{3}) for any 1 < p ≤ 2 and η ∈ L^{q}(ℝ^{3}) for 3/2 < q ≤ 6 by the Sobolev embedding theorem. Since (??) holds a.e. in ℝ^{3}, \overline{u} ∈ H^{1}(ℝ^{3}, S^{3}).

Finally, we calculate (??). Note that u and \overline{u} satisfy (??). As before, if ξ is a 1-form on S^{3} such that dξ = H^{#}(ω_{S^{2}}) and δξ = 0, then \overline{η} = \overline{u}^{#}(ξ) is a 1-form on ℝ^{3} with \overline{η} ∈ L^{2}(ℝ^{3}) and d\overline{η} = d(\overline{u}^{#}(ξ)) = \overline{u}^{#}(dξ) = \overline{u}^{#}(H^{#}(ω_{S^{2}})) = u^{#}(ω_{S^{2}}). Besides, δ\overline{η} = δ\overline{u}^{#}(ξ) = \overline{u}^{#}(δξ) = 0. Hence \overline{η} = η.

One should note that any map in H^{1}(ℝ^{3}, S^{3}) can be approximated strongly by a sequence of smooth maps in H^{1}(ℝ^{3}, S^{3}) (see e.g., [?] and [?]).

We therefore conclude that (??) holds. Note that the right-hand side of (??) is simply the integral of J(\overline{u}), the Jacobian of \overline{u}.

Now we apply the result of Esteban–Muller [?] to conclude that the right-hand side of (??) is the topological degree of \overline{u} (which is well defined, see [?]):

Q(u) = deg(\overline{u}) ∈ ℤ.  (3.20)

To summarize, we have proved the following

Theorem 3.2. For any u ∈ X_{2}, the (generalized) Hopf invariant, Q(u), is well defined. It is, in fact, the topological degree of \overline{u} : ℝ^{3} → S^{3} (the lift of u given in (??)). In particular, Q(u) ∈ ℤ.

3.4 The space X

The most natural mapping space for the variational problem involving the energy functional (??) is the following:

X = {u : ℝ^{3} → S^{2} | E(u) < ∞}.  (3.21)

The purpose of this subsection is to establish

Theorem 3.3. X = X_{2}, and hence, for any u ∈ X, there is a well-defined (generalized) Hopf invariant Q(u) ∈ ℤ.

By the definition of spaces X_{2} and X, it suffices to verify that, for each u ∈ X, one has d(u^{#}(ω_{S^{2}})) = 0 in the sense of distributions.

The 3-form d(u^{#}(ω_{S^{2}})) (which can also be viewed as a scalar-valued distribution) is called the Jacobian of the map u : ℝ^{3} → S^{2}, denoted by J(u). By duality, J(u) can also be viewed as a zero-dimensional current in ℝ^{3}. The following lemma is a special case of Theorem 1.1 and Corollary 1.1 in [?]. See also [?] for some more general statements.

Lemma 3.4. Suppose u ∈ H^{1}(ℝ^{3}, S^{2}) and J(u) is a Radon measure. The Jacobian J(u) then is a zero-dimensional integral current of finite mass in ℝ^{3}. More precisely,

J(u) ≡ d(u^{#}(ω_{S^{2}})) = 4π ∑_{s=1}^{ℓ} d_{s}δ_{a_{s}},  (3.22)

for some d_{s} ∈ ℤ and a_{s} ∈ ℝ^{3}, s = 1, 2, ⋯, ℓ.

In order to show d(u^{#}(ω_{S^{2}})) = 0, it suffices to verify that J(u) ∈ L^{1}(ℝ^{3}) because the latter and (??) would yield J(u) = 0.

Since u^{#}(ω_{S^{2}}) = u · du ∧ du, hence it suffices to show that

d(u · du ∧ du) = du · (du ∧ du).  (3.23)

Note that E(u) < ∞ implies that the right-hand side of (??) is in L^{1}(ℝ^{3}) by the Cauchy–Schwartz inequality.

Proof of (??)

Let {u_{i}} ⊂ C^{1}(ℝ^{3}, S^{2}) such that u_{i} ⇀ u (weakly) in H^{1}(ℝ^{3}, S^{2}) as i → ∞. The existence of such u_{i}’s follows easily from [?]. Thus, in the sense of distributions, one has

d(u · du ∧ du) = lim_{i→∞} d(u_{i} · (du ∧ du)).

If we could show that d(u_{i} · (du ∧ du)) = du_{i} · (du ∧ du), then, since du_{i} · (du ∧ du) ⇀ du · (du ∧ du) (because du ∧ du ∈ L^{2}(ℝ^{3}) and du_{i} ⇀ du in L^{2}(ℝ^{3})), we would finish the proof.

Finally, for each i, u_{i} ∈ C^{1}(ℝ^{3}, S^{2}) fixed, we want to show that d(u_{i} · (du ∧ du)) = du_{i} · (du ∧ du). However, the last identity follows easily from the weak continuity of Jacobians. Indeed, by the weak continuity of Jacobians, and since u_{i} is smooth, one has

u_{i} · (du ∧ du) = lim_{j→∞} u_{i} · (du_{j} ∧ du_{j}).

Hence

d(u_{i} · (du ∧ du)) = lim_{j→∞} d(u_{i} · (du_{j} ∧ du_{j})) = lim_{j→∞} du_{i} · (du_{j} ∧ du_{j}) = du_{i} · (du ∧ du),

in the sense of distributions. We therefore completed the proof of (??), and hence also Theorem ??.

4 The Case of Bounded Domains

Let Ω be a bounded, smooth domain in ℝ^{3}, and let

X_{Ω} = {u | u : Ω → S^{2} such that E(u) < ∞ and u |_{∂Ω} is a constant}.

For any u ∈ X_{Ω}, one can define v ∈ X by simply letting v = u in \overline{Ω} and v = u|_{∂Ω} =constant in Ω^{c}. It is easy to see by the discussion in the previous section that each u ∈ X_{Ω} has a well-defined (generalized) Hopf invariant Q(u) given by

Q(u) = \frac{1}{16π^{2}} ∫_{Ω} η ∧ u^{#}(ω_{S^{2}}) ∈ ℤ,  (4.1)

where η is the 1-form on Ω such that

dη = u^{#}(ω_{S^{2}}), δη = 0 in Ω; ı_{∂Ω}^{#}(η) = 0.  (4.2)

Here ı_{∂Ω} is the inclusion of ∂Ω into ℝ^{3}.

The boundary value problem (??) is elliptic in the sense of Agmon–Douglis– Nirenberg [?]. Hence, by elliptic estimates,

∥η∥_{H^{1}(Ω)} + ∥η∥_{L^{6}(Ω)} ≤ C(Ω)\sqrt{E(u)}.  (4.3)

Theorem 4.1. For any d ∈ ℤ, there is a map u_{(d)} ∈ X_{Ω} such that Q(u_{(d)}) = d and that

E(u_{(d)}) = min{E(u) | u ∈ X_{Ω}, Q(u) = d}.

Proof. Let {u_{i}} ⊂ X_{Ω} be an energy-minimizing sequence with Q(u_{i}) = d, i = 1, 2, · · ·. Thus E(u_{i}) ≤ C(d) for all i = 1, 2, · · ·. We may assume (by taking subsequence if needed) that

u_{i} ⇀ u_{(d)} in H^{1}(Ω, S^{2}) with u_{i} |_{∂Ω} = p_{i} ∈ S^{2}, p_{i} → p = u_{(d)} |_{∂Ω}  (4.4)

and

∂_{k}u_{i} ∧ ∂_{ℓ}u_{i} ⇀ ∂_{k}u_{(d)} ∧ ∂_{ℓ}u_{(d)} in L^{2}(Ω)  (4.5)

as i → ∞. It is then clear that

E(u_{(d)}) ≤ lim inf_{i→∞} E(u_{i}).  (4.6)

Let η_{i} satisfy (??) with u replaced by u_{i}. Then, from (??), we may assume η_{i} → η_{(d)} in L^{2}(Ω). Here η_{(d)} satisfies (??) with u replaced by u_{(d)}. Thus

Q(u_{(d)}) = \frac{1}{16π^{2}} ∫_{Ω} η_{(d)} ∧ u_{(d)}^{#}(ω_{S^{2}})
= lim_{i→∞} \frac{1}{16π^{2}} ∫_{Ω} η_{i} ∧ u_{i}^{#}(ω_{S^{2}}) = d.  (4.7)

Note that, here we may also assume that u_{i} → u_{(d)} in L^{2}(Ω) and that u_{i}(x) → u_{(d)}(x) for a.e. x ∈ Ω. Hence u_{i}^{#}(ω_{S^{2}}) ⇀ u_{(d)}^{#}(ω_{S^{2}}) in L^{2}(Ω).

(??) and (??) then imply that u_{(d)} is an energy minimizer in the (homotopy) class {u ∈ X_{Ω} | Q(u) = d}.

5 Energy Growth Lemma

It was proved long ago by Vakulenko and Kapitanski [?] that for any u ∈ B (see §3.1), one has

|Q(u)|^{3/4} ≤ C_{1}^{−1}E(u),  (5.1)

for a positive constant C_{1}. The same proof works also more generally for any u ∈ X as the proof of (??) involves only Gagliado–Nirenberg–Sobolev’s inequalities (a special case of such a class of inequalities is also called Ladyzhenskaya’s inequality), the classical Hölder inequality, and some elementary observations (see [?] and the remark at the end of §6). Here we want to show the following:

Lemma 5.1. (Energy Growth Lemma) There is a universal constant C_{2} such that

E_{d} = inf{E(u) | u ∈ X, Q(u) = d ∈ ℤ} ≤ C_{2}|d|^{3/4}.  (5.2)

Combining (??) and (??), we conclude that

C_{1}|d|^{3/4} ≤ E_{d} ≤ C_{2}|d|^{3/4}, ∀d ∈ ℤ.  (5.3)

To prove this lemma, we first observe the following fact: if u ∈ C^{1}(ℝ^{3}, S^{2}) is such that u(x) =constant for |x| sufficiently large and that v : S^{2} → S^{2} is a smooth map of degree deg(v), then the Hopf invariant of ũ = v ∘ u : ℝ^{3} → S^{2} satisfies

Q(ũ) = (deg(v))^{2}Q(u).  (5.4)

Indeed, let η be a 1-form over ℝ^{3} such that

dη = ũ^{#}(ω_{S^{2}}), δη = 0 in ℝ^{3}; η(x) → 0 as |x| → ∞.

Then

Q(ũ) = \frac{1}{16π^{2}} ∫_{ℝ^{3}} η ∧ ũ^{#}(ω_{S^{2}}).

Since ũ^{#}(ω_{S^{2}}) = u^{#}(v^{#}(ω_{S^{2}})) and v^{#}(ω_{S^{2}}) is a closed 2-form on S^{2} with ∫_{S^{2}} v^{#}(ω_{S^{2}}) = deg(v), one see that ω′ = v^{#}(ω_{S^{2}})/ deg(v) is again a generator of H_{dR}^{2}(S^{2}) = ℤ. We write ω′ = dξ′ for some 1-form ξ′ on S^{2} satisfying δξ′ = 0. Then, as in §3.3, one has η = u^{#}(ξ′) · (deg v) = η′(deg v). Therefore

Q(ũ) = \frac{(deg v)^{2}}{16π^{2}} ∫_{ℝ^{3}} η′ ∧ u^{#}(ω′).

Here \frac{1}{4π} ∫_{S^{2}} ω′ = 1, dη′ = u^{#}(ω′), and the conclusion (??) follows.

Proof of Lemma ??

We first consider the case d = n^{2}, for a nonnegative integer n. We decompose the upper hemisphere S_{+}^{2} as

S_{+}^{2} = ∪_{i=1}^{n}B(i) ∪ D.

Here B(i)’s are mutually disjoint geodesic balls of radius r ≈ 1/\sqrt{n} inside S_{+}^{2}. We define a Lipschitz map v : S^{2} → S^{2} as follows: v(x) = (0, 0, 1) for all x ∈ S^{2}\∪_{i=1}^{n}B(i), and on each B(i), v is such that v|_{∂B(i)} = (0, 0, 1), v(B(i)) covers S^{2} exactly once, and v : B(i) → S^{2} is orientation-preserving. In other words, the degree of the map from B(i) onto S^{2} is exactly 1. We can further require that ∥∇v∥_{L^{∞}(S^{2})} ≤ c\sqrt{n} for a positive constant c independent of n.

It is rather easy to construct a map h : ℝ^{3} → S^{2} such that h is a constant outside the ball B\sqrt{n}, ∥∇h∥_{L^{∞}(ℝ^{3})} ≤ c/\sqrt{n} for a constant c independent of n, and that Q(h) = 1.

Let u = v ∘ h ∈ B. Then (??) gives Q(u) = n^{2} = (deg v)^{2} = d. On the other hand, ∥∇u∥_{L^{∞}(ℝ^{3})} ≤ c^{2} and u(x) is a constant for x outside the ball B\sqrt{n}. Hence

E(u) ≤ C(\sqrt{n})^{3} = C|d|^{3/4}.

For the general case, we have n^{2} ≤ d < (n + 1)^{2} for some nonnegative integer n. We observe that m = d − n^{2} < (n + 1)^{2} − n^{2} = 2n + 1.

Let h_{0} : B_{1} → S^{2} be a smooth map with h_{0}|_{∂B_{1}} = (0, 0, 1) and Q(h_{0}) = 1. Take m points x_{1}, x_{2}, ⋯, x_{m} ∈ ℝ^{3} such that |x_{i}| >> \sqrt{n} and that |x_{i} − x_{j}| >> 1 + \sqrt{n} for all i, j = 1, 2, ⋯, m, i ≠ j. We then define ũ : ℝ^{3} → S^{2} as follows:

ũ(x) = \{ \begin{matrix}u(x) = (v ∘ h)(x), & for x ∈ B\sqrt{n}(0), \\ h_{0}(x − x_{i}), & for x ∈ B_{1}(x_{i}), i = 1, 2, ⋯, m, \\ (0, 0, 1), & otherwise. \end{matrix}

Here u is constructed as in the case d = n^{2} before.

It is obvious that ũ is a Lipschitz map from ℝ^{3} into S^{2} with ũ(x) = (0, 0, 1) for |x| large. Besides,

Q(ũ) = Q(u) + m = n^{2} + m = d.

Moreover, E(ũ) = E(u)+mE(h_{0}) ≤ C_{1}(\sqrt{n})^{3} ≤ C_{2}d^{3/4}. We have thus proved Lemma ?? in the case that d is nonnegative.

For negative d, one simply needs to change orientation.

6 The Case of Entire Space: Part I

Let u ∈ X (see (??)). Then, from the previous two sections, we know that:

(i) there is a 1-form η on ℝ^{3} such that dη = u^{#}(ω_{S^{2}}), δη = 0, and η = 0 at infinity. Since |dη| = |u^{#}(ω_{S^{2}})| ∈ L^{1}(ℝ^{3}) ∩ L^{2}(ℝ^{3}), we have η ∈ L^{p}(ℝ^{3}) for 3/2 < p ≤ 6. Moreover, ∥η∥_{L^{p}(ℝ^{3})} ≤ C(p)∥dη∥_{L^{q}(ℝ^{3})} = C(p)∥u^{#}(ω_{S^{2}})∥_{L^{q}(ℝ^{3})} for p = 3q/(3 − q), 1 < q ≤ 2. In particular,

∥η∥_{L^{3}(ℝ^{3})} ≤ C(3)∥u^{#}(ω_{S^{2}})∥_{L^{3/2}(ℝ^{3})}
≤ C(3)∥∇u∥_{L^{2}(ℝ^{3})}^{2/3}∥u^{#}(ω_{S^{2}})∥_{L^{2}(ℝ^{3})}^{2/3}
≤ C(3)E(u)^{2/3},  (6.1)

and

∥η∥_{L^{2}(ℝ^{3})} ≤ C(2)∥u^{#}(ω_{S^{2}})∥_{L^{6/5}(ℝ^{3})}
≤ C(2)∥∇u∥_{L^{2}(ℝ^{3})}^{4/3}∥u^{#}(ω_{S^{2}})∥_{L^{2}(ℝ^{3})}^{1/3}
≤ C(2)E(u)^{5/6};  (6.2)

(ii) there is a unique lift \overline{u} : ℝ^{3} → S^{3} satisfying (??) a.e. Thus we have

∥∇\overline{u}∥_{L^{2}(ℝ^{3})}^{2} ≤ E(u) + cE(u)^{5/3}.  (6.3)

We now introduce some notation. Let {Q_{i}(R)}_{i=1}^{∞} be a cubic decomposition of ℝ^{3}. That is,

ℝ^{3} = ∪_{i=1}^{∞}Q_{i}(R),

where each Q_{i}(R) is a closed cube of side length R > 0. Moreover, we assume that the interiors of the cubes Q_{i}(R), i = 1, 2, ⋯, are mutually disjoint, and one of the cubes, say Q_{1}(R), contains the origin as its center.

For all a ∈ ℝ^{3}, with |a| ≤ R/4, we let

Q_{i}(R, a) = a + Q_{i}(R)

denote a translation of Q_{i}(R) by a. Of course, {Q_{i}(R, a)} is another decomposition of ℝ^{3}. We also use the notation

Σ_{R} = ∪_{i=1}^{∞}∂Q_{i}(R), a + Σ_{R} = Σ_{R}(a) = ∪_{i=1}^{∞}∂Q_{i}(R, a),

which are simply the unions of 2-dimensional faces of Q_{i}(R)’s and Q_{i}(R, a)’s, respectively.

The following lemma plays a key role in our proof of the existence of energy minimizers for the Faddeev model in the entire space case.

Lemma 6.1. Suppose u ∈ X with the (generalized) Hopf invariant Q(u) = d ∈ ℤ. Then there is a cubic decomposition {Q_{i}(R_{0}, a_{0})} for some large R_{0} > 0 and a point a_{0} ∈ ℝ^{3} with |a_{0}| ≤ R_{0}/4 such that

∫_{Σ_{R_{0}}(a_{0})}(|∇u|^{2} + |η(u)|^{2}) dσ ≤ \frac{C}{R_{0}} ≤ ε_{0}^{2} << 1.  (6.4)

Moreover, for each Q_{i}(R_{0}, a_{0}), there is an integer k_{i} = k(Q_{i}(R_{0}, a_{0})) such that

|d − ∑_{i=1}^{∞} k_{i}| = |\frac{1}{16π^{2}} ∫ η(u) ∧ u^{#}(ω_{S^{2}}) − ∑_{i=1}^{∞} k_{i}|

≤ ∑_{i=1}^{∞} |\frac{1}{16π^{2}} ∫_{Q_{i}(R_{0},a_{0})} η(u) ∧ u^{#}(ω_{S^{2}}) − k_{i}|
≤ C ∫_{Σ_{R_{0}}(a_{0})}(|∇u|^{2} + |η(u)|^{2}) dσ
≤ \frac{C_{3}}{R_{0}} ≤ ε_{0}^{2} << 1.  (6.5)

Here η(u) is the 1-form in the statement (i) above and ε_{0} > 0 is any given number.

Proof. Since E(u) < ∞, we have by (??) and (??) that

∫_{ℝ^{3}}(|∇u|^{2} + |η(u)|^{2}) dx ≤ E(u) + CE(u)^{5/3}.  (6.6)

Applying Fubini’s theorem, we have

∫_{0}^{R/4} da^{1} ∫_{0}^{R/4} da^{2} ∫_{0}^{R/4} da^{3} ∫_{Σ_{R}(a)}(|∇u|^{2} + |η(u)|^{2}) dσ
≤ CR^{2} ∫_{ℝ^{3}}(|∇u|^{2} + |η(u)|^{2}) dx.  (6.7)

Thus the mean value theorem implies that there is a point a_{0} = (a_{0}^{1}, a_{0}^{2}, a_{0}^{3}) ∈ ℝ^{3} with |a_{0}^{j}| ≤ R/4 (j = 1, 2, 3) such that

∫_{Σ_{R}(a_{0})}(|∇u|^{2} + |η(u)|^{2}) dσ ≤ \frac{CR^{2}}{R^{3}} ∫_{ℝ^{3}}(|∇u|^{2} + |η(u)|^{2}) dx
≤ \frac{C_{4}}{R} ,

where C_{4} is a constant depending also on the value E(u). Hence (??) follows from the above by taking R = R_{0} sufficiently large to start with. In principle, how large R_{0} has to be so that C_{4}/R_{0} is smaller than a given positive constant ε_{0}^{2} may also depend on the value of E(u). However, in our proofs below, we shall apply Lemma ?? to an energy minimizing sequence in X with the given Hopf invariant d. Hence, by the Energy Growth Lemma in §5, we have E(u) ≤ C_{2}|d|^{3/4}. Thus the choice of R_{0} will eventually depend only on d and ε_{0}.

We now proceed to show (??). Let Q(R_{0}) be one of the Q_{i}(R_{0}, a_{0})’s in the cubic decomposition of ℝ^{3} and \overline{u} : ℝ^{3} → S^{3} be the lift of u : ℝ^{3} → S^{2} defined by (??). In view of (??), we have |∇\overline{u}|^{2} ≤ |∇u|^{2} + |η(u)|^{2}. Hence, in particular,

∫_{∂Q(R_{0})} |∇\overline{u}|^{2} dσ ≤ ∫_{∂Q(R_{0})}(|∇u|^{2} + |η(u)|^{2}) dσ
≤ ε_{0}^{2} << 1.  (6.8)

Let Q(2R_{0}) be the cube which has the side length 2R_{0} and the same center as Q(R_{0}). We consider the following problem:

Δv = 0 in B_{2R_{0}} \ B_{R_{0}};
v = \overline{u} ∘ F on ∂B_{R_{0}}, v = \underline{u} on ∂B_{2R_{0}},  (6.9)

where F : B_{2R_{0}} \ B_{R_{0}} → Q(2R_{0}) \ Q(R_{0}) is a bi-Lipschitz map and

\underline{u} = − ∫ _{∂B_{R_{0}}} \overline{u} ∘ F /|− ∫ _{∂B_{R_{0}}} \overline{u} ∘ F | ∈ S^{3}.

Note that, by the Poincaré inequality,

− ∫ _{∂B_{R_{0}}} |\overline{u} ∘ F − − ∫ _{∂B_{R_{0}}} \overline{u} ∘ F |^{2} ≤ c_{F} ∫_{∂Q(R_{0})} |∇\overline{u}|^{2} dσ
≤ ε_{0}^{2} << 1.

Thus, the distance

dist(− ∫ _{∂B_{R_{0}}} \overline{u} ∘ F, S^{3}) = |\underline{u} − − ∫ _{∂B_{R_{0}}} \overline{u} ∘ F | ≤ ε_{0} << 1.

By elliptic estimates, one has v ∈ H^{3/2}(B_{2R_{0}} \ B_{R_{0}}) and

∥v − \underline{u}∥_{H^{3/2}(B_{2R_{0}}\B_{R_{0}})} ≤ C∥∇_{T} v∥_{L^{2}(∂Q(R_{0}))},  (6.10)

where C is an absolute constant independent of R_{0} due to the scaling invariance of the quantities on both sides of (??). That is, one has (??) for the case R_{0} = 1 and the general case follows from a rescaling.

Hence the Sobolev embedding yields

∥∇v∥_{L^{3}(B_{2R_{0}}\B_{R_{0}})} ≤ C∥v − \underline{u}∥_{H^{3/2}(B_{2R_{0}}\B_{R_{0}})}
≤ C∥∇\overline{u}∥_{L^{2}(∂Q(R_{0}))} ≤ ε_{0} << 1.  (6.11)

Next, we want to show that if ε_{0} is sufficiently small, then

sup_{s∈B_{2R_{0}}\B_{R_{0}}} dist(v(x), S^{3})

is also small. Indeed, the boundary data of v is close to \underline{u} ∈ S^{3} (on ∂B_{2R_{0}}, the boundary value is \underline{u}) because

− ∫ _{∂B_{R_{0}}}|\overline{u} ∘ F − \underline{u}|^{2}
≤ 2 − ∫ _{∂B_{R_{0}}} |\overline{u} ∘ F − − ∫ _{∂B_{R_{0}}} \overline{u} ∘ F |^{2} + 2|− ∫ _{∂B_{R_{0}}} \overline{u} ∘ F − \underline{u}|^{2}
≤ Cε_{0}^{2}.

This along with (??) imply that

dist(v(x), S^{3}) ≤ |v(x) − \underline{u}| ≤ C(δ)ε_{0}  (6.12)

(which is small for small but fixed δ) for all x ∈ B_{2R_{0}} \ B_{R_{0}} such that

dist(x, ∂(B_{2R_{0}} \ B_{R_{0}})) ≥ δR_{0}.

Here δ ∈ (0, 1) is suitably small.

Due to the fact that all estimates are scaling invariant under the natural rescaling v_{R_{0}}(y) = v(R_{0}y), y ∈ B_{2} \ B_{1}, we may simply assume R_{0} = 1.

By continuity of v_{R_{0}} near ∂B_{2}, one sees that it suffices to verify that dist(v_{R_{0}}(y), S^{3}) is small for all y such that dist(y, ∂B_{1}) ≤ δ. Since

v_{R_{0}}(y) = ∫_{∂(B_{2}\B_{1})} P (y, ξ)f(ξ) dσ(ξ).

where f(ξ) = \overline{u}(R_{0}ξ) for ξ ∈ ∂B_{1} and f(ξ) = \underline{u} for ξ ∈ ∂B_{2} and P is the Poisson kernel.

For any y ∈ B_{2} \ B_{1} with dist(y, ∂B_{1}) ≤ δ, we let \overline{y} ∈ ∂B_{1} so that dist(y, ∂B_{1}) = |y − \overline{y}|. We take m sufficiently large (which will be chosen later). By the Poincaré inequality, one sees, for any 0 < r ≤ 1/2, that

\overline{f}_{r}(\overline{y}) = ∫_{\{ξ | |ξ−\overline{y}|≤r\}∩∂B_{1}} f(ξ) dσ(ξ)/|\{ξ | |ξ − \overline{y}| ≤ r\} ∩ ∂B_{1}|

satisfies

dist^{2}(\overline{f}_{r}, S^{3}) ≤ C_{0}ε_{0}^{2} << 1.  (6.13)

We want to show that |v_{R_{0}}(y) − \overline{f}_{mr}(\overline{y})| is small. Here dist(y, ∂B_{1}) = r ≤ δ. This last fact along with (??) and (??) will yield that sup{dist(v(x), S^{3}) | x ∈ B_{2R_{0}} \ B_{R_{0}}} is small.

To show that |v_{R_{0}}(y) − \overline{f}_{mr}(\overline{y})| is small, we estimate the Poisson integral directly. First of all,

| ∫_{∂(B_{2}\B_{1})∩\{ξ | |ξ−\overline{y}|≥mr\}} P (y, ξ)(f(ξ) − \overline{f}_{mr}(\overline{y})) dσ(ξ)| (|f(ξ)| ≤ 1)
≤ 2 ∫_{∂(B_{2}\B_{1})∩\{ξ | |ξ−\overline{y}|≥mr\}} P (y, ξ) dσ(ξ) = O(\frac{1}{m}),  (6.14)

which is small if one takes m suitably large. Secondly,

| ∫_{∂(B_{2}\B_{1})∩\{ξ | |ξ−\overline{y}|≤mr\}} P (y, ξ)(f(ξ) − \overline{f}_{mr}(\overline{y})) dσ(ξ)|
≤ \frac{C}{r^{2}} ∫_{∂(B_{2}\B_{1})∩\{ξ | |ξ−\overline{y}|≤mr\}} |f(ξ) − \overline{f}_{mr}(\overline{y})| dσ(ξ)
≤ Cm^{2} ∫_{∂(B_{2}\B_{1})∩\{ξ | |ξ−\overline{y}|≤mr\}} |∇\overline{u}|^{2} dσ(ξ) ≤ Cm^{2}ε_{0}^{2}.  (6.15)

The latter again can be made small if ε_{0} is sufficiently small.

Since v(x) is close to S^{3} for all x ∈ B_{2R_{0}} \ B_{R_{0}}, we can define \overline{u}_{1}(x) as follows:

\overline{u}_{1}(x) = \{ \begin{matrix}\overline{u}(x) & for x ∈ B_{R_{0}}, \\ π_{S^{3}}(v(x)) & for x ∈ B_{2R_{0}} \ B_{R_{0}}, \end{matrix}  (6.16)

where π_{S^{3}} is the nearest point projection onto S^{3}.

It is easy to see that

∥∇π_{S^{3}}(v(x))∥_{L^{3}(B_{2R_{0}}\B_{R_{0}})} ≤ 2∥∇v∥_{L^{3}(B_{2R_{0}}\B_{R_{0}})}
≤ C∥∇\overline{u}∥_{L^{2}(∂Q(R_{0}))} ≤ ε_{0} << 1.  (6.17)

Moreover, the Hopf invariant of the map u_{1} ∈ X,

u_{1} = \{ \begin{matrix}H ∘ \overline{u}_{1} & on B_{2R_{0}}, \\ \underline{u} & outside B_{2R_{0}} \end{matrix}

is an integer denoted by k(Q(R_{0})) ∈ ℤ, which is exactly the degree of the map \overline{u}_{1} ∘ F^{−1} : Q(2R_{0}) → S^{3}.

A direct calculation then implies that

|\frac{1}{16π^{2}} ∫_{Q(R_{0})} η(u) ∧ u^{#}(ω_{S^{2}}) − k(Q(R_{0}))|

≤ |\frac{1}{16π^{2}} ∫_{Q(2R_{0})\Q(R_{0})}(\overline{u}_{1} ∘ F^{−1})^{#}(ξ) ∧ (\overline{u}_{1} ∘ F^{−1})^{#}(H^{#}(ω_{S^{2}}))|
≤ C ∫_{Q(2R_{0})\Q(R_{0})} |∇(\overline{u}_{1} ∘ F^{−1})|^{3}(x) dx
≤ C( ∫_{∂Q(R_{0})} |∇\overline{u}|^{2} dσ)^{3/2}
≤ Cε_{0} ∫_{∂Q(R_{0})} |∇\overline{u}|^{2} dσ.

The second conclusion of Lemma ??, (??), follows from the last inequality. Note that C_{3} ≤ Cε_{0}.

We now go back to the fundamental problem

E_{d} ≡ inf{E(u) | u ∈ X, Q(u) = d}.  (6.18)

Since d = 0 is trivial (the infimum is realized by constant maps), we shall simply assume d ≠ 0, d ∈ ℤ. Our main result is the following

Theorem 6.2. Let {u_{i}} be an energy-minimizing sequence of the problem (??). Then, there are a subsequence which is still denoted by {u_{i}} for convenience, a decomposition

d = d_{1} + d_{2} + ⋯ + d_{ℓ}, 1 ≤ ℓ ≤ c_{1}|d|^{3/4},

and a sequence of points x_{i,1}, x_{i,2}, ⋯, x_{i,ℓ} ∈ ℝ^{3} where d_{1}, d_{2}, ⋯, d_{ℓ} ∈ ℤ such that v_{i,s}(x) = u_{i}(x−x_{i,s}) converges weakly in the metric d_{E} to some v_{s} ∈ X with Q(v_{s}) = d_{s} as i → ∞. Besides, |x_{i,s} − x_{i,t}| → ∞ as i → ∞ for s, t = 1, 2, ⋯, ℓ and s ≠ t.

Proof. Let {u_{i}} be an energy-minimizing sequence for (??) so that E(u_{i}) → E_{d} as i → ∞. We apply Lemma ?? to each u_{i} (with the same R_{0} and a sequence {a_{i}} ⊂ ℝ^{3} satisfying |a_{i}| ≤ R_{0}/4 for all i). For fixed i, we observe that

∑_{j=1}^{∞} ∫_{Q_{j}(R_{0},a_{i})} |η(u_{i})||u_{i}^{#}(ω_{S^{2}})| dx ≤ ∥η(u_{i})∥_{L^{2}(ℝ^{3})}∥u_{i}^{#}(ω_{S^{2}})∥_{L^{2}(ℝ^{3})} (by (??))
≤ CE(u_{i})^{5/6}E(u_{i})^{1/2} = CE(u_{i})^{4/3}.  (6.19)

Suppose that, for fixed i, there are at least m quantities k(Q_{j}(R_{0}, a_{i})) satisfying k(Q_{j}(R_{0}, a_{i})) = k_{j}^{i} ≠ 0 when j = j_{1}, j_{2}, ⋯, j_{m}. Then

∑_{j=1}^{∞} |\frac{1}{16π^{2}} ∫_{Q_{j}(R_{0},a_{i})} η(u_{i}) ∧ u_{i}^{#}(ω_{S^{2}})|

≥ ∑_{j=1}^{∞} |k_{j}^{i}| − ∑_{j=1}^{∞} |\frac{1}{16π^{2}} ∫_{Q_{j}(R_{0},a_{i})} η(u_{i}) ∧ u_{i}^{#}(ω_{S^{2}}) − k_{j}^{i}|
≥ m − ε_{0}^{2} (see (??) of Lemma ??).  (6.20)

On the other hand, the left-hand side of (??) is bounded above by (see (??)) CE(u_{i})^{4/3}/16π^{2} ≤ C|d|. Thus m ≤ C|d|/16π^{2}. We therefore may assume, by taking subsequence if needed, that for this energy-minimizing sequence, there are always exactly m of the k_{j}^{i}’s, j = 1, 2, ⋯, which are nonzero for each u_{i}. For convenience, we name those corresponding cubes Q_{j}(R_{0}, a_{i}) (with nonzero k_{j}^{i}’s) by Q_{i}^{1}, Q_{i}^{2}, ⋯, Q_{i}^{m}.

We introduce the following subsets, S_{1}, S_{2}, ⋯, S_{ℓ} of the set S = {1, 2, ⋯, m} inductively. Let s_{1} = 1. We choose a subsequence of {u_{i}}, which we again denote by {u_{i}}, such that there is a subset S_{1} of S with s_{1} ∈ S_{1} and

lim_{i→∞} dist(Q_{i}^{s_{1}}, Q_{i}^{k})

exists (finite) for each s ∈ S_{1}. Moreover, for any k ∈ S \ S_{1}, we have

lim_{i→∞} dist(Q_{i}^{s_{1}}, Q_{i}^{k}) = ∞.

The existence of such a subsequence of {u_{i}} can be verified in less than m steps of choosing suitable subsequences.

If S \ S_{1} is nonempty, then we set s_{2} = min{k ∈ S \ S_{1}} and repeat the above procedure to find a subsequence of {u_{i}} such that there is a subset S_{2} of S \ S_{1} so that

lim_{i→∞} dist(Q_{i}^{s_{2}}, Q_{i}^{k})

exists (finite) for each k ∈ S_{1}, and for any k ∈ S \ (S_{1} ∪ S_{2}), there holds

lim_{i→∞} dist(Q_{i}^{s_{2}}, Q_{i}^{k}) = ∞.

We can continue in the way described above. Thus we are able to find ℓ numbers 1 = s_{1} < s_{2} < ⋯ < s_{ℓ}, the corresponding subsets S_{1}, S_{2}, ⋯, S_{ℓ} of S, and a subsequence of {u_{i}} (denoted by itself for simplicity) with the following properties

S = {1, 2, ⋯, m} = S_{1} ∪ S_{2} ∪ ⋯ ∪ S_{ℓ},
S_{s} ∩ S_{t} = ∅ for s ≠ t, s, t = 1, 2, ⋯, ℓ;
s_{1} = 1,
S_{1} = {k ∈ S | lim_{i→∞} dist(Q_{i}^{s_{1}}, Q_{i}^{k}) < ∞};
s_{2} ∈ S \ S_{1},
S_{2} = {k ∈ S | lim_{i→∞} dist(Q_{i}^{s_{2}}, Q_{i}^{k}) < ∞};
⋯  · · · · · ·
s_{ℓ} ∈ S \ ∪_{s=1}^{ℓ−1}S_{s},
S_{ℓ} = {k ∈ S | lim_{i→∞} dist(Q_{i}^{s_{ℓ}}, Q_{i}^{k}) < ∞}.

We let x_{i,1}, x_{i,2}, ⋯, x_{i,ℓ} be the centers of the cubes Q_{i}^{s_{1}}, Q_{i}^{s_{2}}, ⋯, Q_{i}^{s_{ℓ}}, denoted thereafter as Q_{i,1}, Q_{i,2}, ⋯, Q_{i,ℓ}, and let v_{i,s}(x) = u_{i}(x − x_{i,s}), s = 1, 2, ⋯, ℓ. We also set

M_{s} = max { lim_{i→∞} dist(Q_{i,s}, Q_{i,t}) | t ∈ S_{s}}, s = 1, 2, ⋯, ℓ.

Then |x_{i,s} − x_{i,t}| → ∞ as i → ∞ for s ≠ t and s, t = 1, 2, ⋯, ℓ.

Finally, by further taking subsequences if necessary, we see that

d_{s} ≡ lim_{i→∞} ∑_{t∈S_{s}} k(Q_{i,t})

exists for s = 1, 2, ⋯, ℓ and that v_{i,s} converges to v_{s} weakly in the metric d_{E} for each s = 1, 2, ⋯, ℓ as i → ∞.

By Lemma ?? (see (??)), we see that

∑_{s=1}^{ℓ} ∑_{t∈S_{s}} k(Q_{i,t}) = d

for each i. Thus d = ∑_{s=1}^{ℓ} d_{s} follows.

To complete the proof of Theorem ??, we need the following lemma.

Lemma 6.3. The Hopf invariant of v_{s}, Q(v_{s}), is exactly d_{s}.

Proof. We may prove the lemma under the additional assumption that x_{i,s} = 0 otherwise we simply consider maps and cubes translated by x_{i,s} in the sequence.

For all n sufficiently large so that nR_{0} > 2M_{s}, we let

Ω_{n}^{i} = ∪{Q_{j}(R_{0}, a_{i}) | dist(Q_{j}(R_{0}, a_{i}), Q_{i,s}) ≤ nR_{0}, j = 1, 2, ⋯}.

Then Ω_{n}^{i} is a sequence of bounded Lipschitz domains for i = 1, 2, · · ·. Moreover, Ω_{n}^{i} → Ω_{n} as i → ∞ with

Ω_{n} = ∪{Q_{j}(R_{0}, a_{∞}) | dist(Q_{j}(R_{0}, a_{∞}), Q_{∞,s}) ≤ nR_{0}, j = 1, 2, ⋯},

where a_{∞} = lim_{i→∞} a_{i} (we can assume that the limit exists by choosing a suitable subsequence if necessary) and Q_{∞,s} = lim_{i→∞} Q_{i,s} (note that since we assume x_{i,s} = 0, the centers of Q_{∞,s} and Q_{i,s} are all at the origin).

In view of our discussion in §3 and §4, it is easy to see, for each fixed j, that

\frac{1}{16π^{2}} ∫_{Q_{j}(R_{0},a_{i})} η(u_{i}) ∧ u_{i}^{#}(ω_{S^{2}}) → \frac{1}{16π^{2}} ∫_{Q_{j}(R_{0},a_{∞})} η(v_{s}) ∧ v_{s}^{#}(ω_{S^{2}})

as i → ∞ (note that we have assumed that x_{i,s} = 0 so that u_{i} converges weakly to v_{s} in the metric d_{E}).

One then deduces from Lemma ?? (see (??)) that, for each such n sufficiently large that nR_{0} > 2M_{s}, one has

|\frac{1}{16π^{2}} ∫_{Ω_{n}} η(v_{s}) ∧ v_{s}^{#}(ω_{S^{2}}) − d_{s}| << 1.  (6.21)

Since v_{s} ∈ X and

Q(v_{s}) = \frac{1}{16π^{2}} ∫_{ℝ^{3}} η(v_{s}) ∧ v_{s}^{#}(ω_{S^{2}}) ∈ ℤ,

we see by letting n → ∞ in (??) that Q(v_{s}) = d_{s}.

Finally, we claim that each v_{s} has a positive energy, E(v_{s}) ≥ c_{0} > 0, s = 1, 2, ⋯, ℓ. Indeed, since

|\frac{1}{16π^{2}} ∫_{Q_{i}^{s_{j}}} η(u_{i}) ∧ u_{i}^{#}(ω_{S^{2}}) − k_{i}^{j}| << 1

for some nonzero integer k_{i}^{j}, hence, after suitable translations, we have

|\frac{1}{16π^{2}} ∫_{Q(R_{0},0)} η(v_{s}) ∧ v_{s}^{#}(ω_{S^{2}})| ≥ \frac{1}{2}, ∀s,

where R_{0} > 0 is sufficiently large. Thus, similar to (??), we have

E(v_{s})^{4/3} ≥ C ∫_{Q(R_{0},0)} |η(v_{s})| |v_{s}^{#}(ω_{S^{2}})| dx
≥ C| ∫_{Q(R_{0},0)} η(v_{s}) ∧ v_{s}^{#}(ω_{S^{2}})| ≥ 8π^{2}C, ∀s.

Remark. The estimate (??) is actually a refinement of the topological lower bound (??) for maps in X. In fact, replacing the integrals on the left-hand side of (??) by a single integral over the full space ℝ^{3}, we have

16π^{2}|Q(u)| = | ∫_{ℝ^{3}} η(u) ∧ u^{#}(ω_{S^{2}})| ≤ ∫_{ℝ^{3}} |η(u)| |u^{#}(ω_{S^{2}})| dx
≤ ∥η(u)∥_{L^{2}(ℝ^{3})} ∥u^{#}(ω_{S^{2}})∥_{L^{2}(ℝ^{3})} ≤ CE(u)^{5/6}E(u)^{1/2}
= CE(u)^{4/3},

which is exactly (??).

7 The Case of Entire Space: Part II

In this section, we present a few extensions of our analysis in §6. The results obtained here complete the proofs of the rest of statements in the theorems in §2. As a byproduct, we also obtain a proof for the existence of the degree one Skyrme solitons.

7.1 The existence of a least positive-energy minimizer and the existence of energy minimizers realizing infinitely many values of the Hopf invariant

The energy lower bound (??) or (??) implies that there is d_{0} ∈ ℤ \ {0} such that

E_{d_{0}} = min{E_{d} | d ∈ ℤ \ {0}}.  (7.1)

For this integer d_{0}, we have

Theorem 7.1. The problem

inf{E(u) | u ∈ X, Q(u) = d_{0}} = E_{d_{0}}  (7.2)

can be solved by some u_{0} ∈ X.

Proof. Since E_{d} → ∞ as |d| → ∞, we see that (??) makes sense.

For this d_{0}, we consider the energy-minimizing sequence {u_{i}} for the problem (??). By Theorem ??, we have a decomposition d_{0} = d_{1} + ⋯ + d_{ℓ} with 1 ≤ |ℓ| ≤ c_{1}|d_{0}|^{3/4} and maps v_{1}, ⋯, v_{ℓ} ∈ X with Q(v_{s}) = d_{s}, d_{s} ≠ 0, s = 1, ⋯, ℓ.

Moreover, by the weak lower semicontinuity of E(·), it is seen from the proof of Theorem ?? that

E_{d_{0}} = lim inf_{i→∞} E(u_{i}) ≥ ∑_{s=1}^{ℓ} E(v_{s}) ≥ ∑_{s=1}^{ℓ} E_{d_{s}} ≥ ℓE_{d_{0}}.  (7.3)

Since d_{s} ≠ 0 for s = 1, ⋯, ℓ, we have ℓ = 1 and d_{0} = d_{1}. In other words, for d = d_{0}, the energy-minimizing sequence {u_{i}}, after a suitable translation, converges weakly to v_{1} such that Q(v_{1}) = d_{0} and E(v_{1}) = E_{d_{0}}. In fact, {u_{i}} (after translations) converges strongly to v_{1} in d_{E} as i → ∞. We therefore complete the proof of the theorem.

Theorem 7.2. There is an infinite subset 𝕊 of ℤ such that for any d ∈ 𝕊 the problem

E_{d} = inf{E(u) | u ∈ X, Q(u) = d}  (7.4)

has a solution.

Proof. By Theorem ??, we see that, if we define

𝕊 = {d ∈ ℤ | the problem (??) has a solution},

then 𝕊 ≠ ∅ because d_{0} ∈ 𝕊. We need to show that 𝕊 is infinite.

Otherwise assume that 𝕊 is finite. Set d^{0} = max{d ∈ 𝕊}. Then d_{0} ≤ d^{0}.

We first repeat part of the proof of Theorem ??. For any d ≥ 1, consider the energy-minimizing sequence {u_{i}} for the problem (??). There is a decomposition d = d_{1} + ⋯ + d_{ℓ} and maps v_{1}, ⋯, v_{ℓ} ∈ X with Q(v_{s}) = d_{s}, d_{s} ≠ 0, s = 1, ⋯, ℓ such that

E_{d} = lim inf_{i→∞} E(u_{i}) ≥ ∑_{s=1}^{ℓ} E(v_{s}) ≥ ∑_{s=1}^{ℓ} E_{d_{s}}.  (7.5)

Next, if d_{s} ∉ 𝕊, we may repeat the above decomposition with d = d_{s}. Since E_{d} is finite, after finitely many steps, we will have to stop with the following results:

There is an index decomposition

d = d′_{1} + d′_{2} + ⋯ + d′_{ℓ′};  (7.6)

there holds the subadditive energy decomposition

E_{d} ≥ E_{d′_{1}} + E_{d′_{2}} + ⋯ + E_{d′_{ℓ′}};  (7.7)

and, the integers d′_{1}, d′_{2}, ⋯, d′_{ℓ′} all belong to 𝕊, hence

|d′_{1}|, |d′_{2}|, ⋯, |d′_{ℓ′}| ≤ d^{0}.  (7.8)

Combining (??) and (??), we have d ≤ ℓ′d^{0}; combining (??) with the definition of d_{0}, we have E_{d} ≥ ℓ′E_{d_{0}}. Consequently, E_{d} ≥ (E_{d_{0}}/d^{0})d, which contradicts (??) (Energy Growth Lemma) when d is sufficiently large.

7.2 An existence theorem for the Skyrme model

First, we remark that our method may be applied to the Skyrme model to obtain an existence theory parallel to that for the Faddeev model presented above.

Recall that, in the static case, the Skyrme energy has the form

E(n) = ∫_{ℝ^{3}} { ∑_{1≤k≤3} |∂_{k}n|^{2} + ∑_{1≤k<ℓ≤3} |∂_{k}n ∧ ∂_{ℓ}n|^{2}} dx,  (7.9)

where n maps ℝ^{3} into S^{3}. Hence, similar to the Faddeev model, the relevant topological invariant is the Brouwer degree (when n is viewed as a map from S^{3} = ℝ^{3} ∪ {∞} into itself) which may be represented as an integral as well,

deg(n) = \frac{1}{2π^{2}} ∫_{ℝ^{3}} det(n, ∇n) dx.  (7.10)

For any integer d, one is interested in the solvability of the optimization problem

E_{d} = inf{E(n) | E(n) < ∞, deg(n) = d}.  (7.11)

It can be shown that all the foregoing analysis (except the (3/4)-growth law) may be carried out for the problem (??). In particular, we have

Theorem 7.3. For d = ±1, the problem (??) has a solution.

Proof. First we note that the problems with d = 1 and d = −1 are equivalent because their solutions only differ by a choice of orientation of S^{3}. In particular, E_{1} = E_{−1}.

Recall that, as in the Faddeev model case, we can obtain a decomposition 1 = d_{1} + ⋯ + d_{ℓ}, d_{s} ∈ ℤ, s = 1, ⋯, ℓ, so that

E_{1} ≥ E_{d_{1}} + ⋯ + E_{d_{ℓ}}.  (7.12)

In order to prove the stated conclusion, it suffices to show that this decomposition is trivial. That is, ℓ = 1 and d_{1} = ±1. In fact, if this is not trivial, then there is a d_{s} (1 ≤ s ≤ ℓ) so that |d_{s}| ≥ 2. Hence (??) says that

E_{1} > E_{d_{s}}.  (7.13)

On the other hand, however, Esteban [?] has obtained the estimates

E_{d} ≥ 6|S^{3}||d|, ∀d ∈ ℤ; E_{1} ≤ 6\sqrt{2}|S^{3}|,  (7.14)

where |S^{3}| is the volume of the unit 3-sphere. Inserting (??) into (??), we arrive at a contradiction.

Note. Theorem ?? was stated in [?] but the proof seems to be incomplete. Our proof fills this gap from a different approach.

In fact, (??) implies that

E_{±1} = min{E_{d} | d ∈ ℤ \ {0}}.

Hence, since Theorem ?? holds also for the Skyrme model, Theorem ?? follows naturally.

7.3 Implication of a technical lemma

We now show that the analysis of §6 can be strengthened in view of the following technical lemma.

Lemma 7.4. Let R ≥ 1 be a constant and u : B_{2R} \ B_{R} → S^{2} be such that

E(u, B_{2R} \ B_{R}) = ∫_{B_{2R}\B_{R}} {|∇u|^{2} + ∑_{1≤k<ℓ≤3} |∂_{k}u ∧ ∂_{ℓ}u|^{2}} dx ≤ ε_{0}  (7.15)

for some suitably small number ε_{0} > 0. Then, after making a global rotation if necessary, there is a map ũ : B_{2R} \ B_{R} → S^{2} such that

(i) ũ = u on ∂B_{R}.

(ii) ũ = (0, 0, 1) on ∂B_{2R}.

(iii) E(ũ, B_{2R} \ B_{R}) ≤ C_{0}E(u, B_{2R} \ B_{R}) ≤ C_{0}ε_{0} where C_{0} > 0 is a constant independent of R, ε_{0}, u, and ũ.

This lemma is presented in Appendix 2 in Esteban [?] with an incomplete proof. In view of the validity of Lemma ??, we have

Theorem 7.5. Each v_{s} obtained in Theorem ?? is a solution to the minimization problem

inf{E(u) | u ∈ X, Q(u) = d_{s}}, s = 1, 2, ⋯, ℓ.

In particular, each d_{s} ≠ 0 (s = 1, 2, ⋯, ℓ) and

E_{d} = E_{d_{1}} + E_{d_{2}} + ⋯ + E_{d_{ℓ}}

holds.

Proof. By Theorem ??, we have a decomposition d = d_{1} + d_{2} + ⋯ + d_{ℓ} with 1 ≤ ℓ ≤ C_{1}|d|^{3/4} and maps v_{1}, v_{2}, ⋯, v_{ℓ} ∈ X with Q(v_{s}) = d_{s} (s = 1, 2, ⋯, ℓ).

By the weak lower semicontinuity of E(·) and the definition of d, we have

∑_{s=1}^{ℓ} E_{d_{s}} ≤ ∑_{s=1}^{ℓ} E(v_{s}) ≤ lim inf_{i→∞} E(u_{i}) = E_{d},  (7.16)

where {u_{i}} is the minimizing sequence of (??) which generates the maps v_{1}, v_{2}, ⋯, v_{ℓ} by Theorem ??.

On the other hand, for each v_{s}, one may find a large R_{s} ≥ 1 so that E(v_{s}, B_{2R_{s}} \ B_{R_{s}}) ≤ ε_{s} ≤ ε_{0}. By Lemma ??, we may find a map ṽ_{s} ∈ X such that ṽ_{s} = v_{s} on B_{R_{s}} and that ṽ_{s} =constant outside B_{2R_{s}} with E(ṽ_{s}) ≤ E(v_{s}) + C_{0}ε_{s}. It is rather easy to see that Q(v_{s}) = d_{s}. Thus we can now glue ṽ_{s}’s together (by translations) to find a new map ũ with Q(ũ) = ∑_{s=1}^{ℓ} d_{s} = d and

ũ(x) = \{ \begin{matrix}ṽ_{s}(x − x_{s}), & for x ∈ B_{2R_{s}}(x_{s}), \\ constant, & otherwise, \end{matrix}

E_{d} ≤ E(ũ) ≤ ∑_{s=1}^{ℓ} E(v_{s}) + C_{0} ∑_{s=1}^{ℓ} ε_{s} ≤ E_{d} + C_{0}ℓε_{0}.  (7.17)

Here ε_{s} > 0 can be made arbitrarily small if one is willing to take R_{s} sufficiently large and the balls B_{2R_{s}}(x_{s}) (s = 1, 2, ⋯, ℓ) are nonoverlapping. Therefore (??) and (??) imply that

∑_{s=1}^{ℓ} E(v_{s}) = E_{d}.  (7.18)

We can show that each v_{s} must be an energy minimizer with Q(v_{s}) = d_{s}. Indeed, if v_{s_{0}} is not a minimizer, then there is an R_{0} > 0 and v ∈ X such that Q(v) = Q(v_{s_{0}}) and

E(v, B_{R}) ≤ E(v_{s_{0}}, B_{R}) − δ_{0}, R ≥ R_{0},

for some δ_{0} > 0. Then taking R_{s} >> R_{0} (s = 1, 2, ⋯, ℓ) and using the same gluing method as in the proof of Theorem ??, we can find a new map ũ (with v_{s_{0}} being replaced by v) so that Q(ũ) = d and

E(ũ) ≤ E_{d} + ε − δ_{0},  (7.19)

where ε > 0 can be chosen to be arbitrarily small. It is clear that (??) contradicts the definition of E_{d} because E_{d} ≤ E(ũ).

Finally, since each v_{s} is an energy minimizer in the class {u ∈ X | Q(u) = d_{s}} and E(v_{s}) > 0, one must have d_{s} ≠ 0. Moreover, since v_{s}’s are energy minimizers, by (??) we have E_{d} = ∑_{s=1}^{ℓ} E_{d_{s}} as expected.

8 Conclusion

In conclusion, we have developed an existence theory for Faddeev’s knots characterized by the Hopf invariant, Q, and realized as stable solitons minimizing the Faddeev energy E. The sublinear asymptotic estimate E ∼ |Q|^{3/4} implies that, at high charges, a clustered (knotted soliton) structure is preferred over a separated (multisoliton) structure. Such a property is in sharp contrast with the conventional linear growth result E ∼ |Q| obtained for other important (classical) quantum field theory models [?, ?, ?, ?, ?, ?, ?] for which the existence of multisolitons (with widely separated energy lumps) is a common phenomenon. Besides, the analysis of this paper shows also that the sublinear growth law and the existence of one nontrivial energy minimizer imply in an essential way the existence of energy minimizers among infinitely many topological classes.

Acknowledgments. Fanghua Lin was supported in part by NSF grant DMS– 9896391 and ChangJiang Professorship Fund at the Zhejiang University.Part of the research was carried out while the first author was visiting the Center for Mathematical Sciences at the Zhejiang University.He would like to thank the warm hospitality of the Center. Yisong Yang was supported in part by NSF grants DMS–9972300 and DMS–9729992 through IAS. It is a pleasure to thank Roman Jackiw and Edward Miller for some helpful comments.

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